A slight confusion regarding dark energy & energy conservation

[haisydinh]

I know there has been a lot of threads on the forum about this topic, but my question is slightly different from the others. I have recently read this article by Sean Carroll (http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/); now, I'm completely aware (& convinced) that energy is not conserved in GR, precisely because of the reasons mentioned in that article. However, in that same article, Sean admitted that it is not incorrect to think that energy is conserved in GR, and i'd like to understand why he could say so. Here is the quote:

In particular, a lot of folks would want to say “energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on.” Which seems pretty sensible at face value. There’s nothing incorrect about that way of thinking about it; it’s a choice that one can make or not, as long as you’re clear on what your definitions are.

Now, what does he mean exactly by this quote? Re-reading this several time, my understanding is this: Although the total dark energy increases as the universe expands, this does not violate the energy conservation. This is because the increase in dark energy also causes more curvature in space-time (due to GR), which consequently increases the gravitational field. And since the gravitational field stores negative energy, it cancels out the positive energy of the dark energy.
Is this the correct description of what actually is going on? Or am I just misinterpreting Sean's article? I mean this explanation sounds a bit weird to me.

Thank you very much in advance! :)

 

[PWiz]

The problem is that energy is frame variant. The law of conservation of energy applies locally in spacetime, but you can't generalize it for the entire universe.

 

[haisydinh]

The problem is that energy is frame variant. The law of conservation of energy applies locally in spacetime, but you can't generalize it for the entire universe.

Can you please elaborate a bit? At least from what I currently understand, the quote above of Sean Carroll basically says that it's ok to claim that the law of energy conservation holds in general relativity, and thus the conservation law also holds for the case of dark energy. But I don't understand why he can make this claim.

Another quote from the article says that: "there’s energy in the gravitational field, but it’s negative, so it exactly cancels the energy you think is being gained in the matter fields". This is where I don't understand. Gravitational field of what? How does that cancel out the energy of the new dark energy that is put into the universe as the universe expands?

Thank you very much!

 

[PWiz]

I'm not 100% sure here, but this article might help give some additional references on the "negative g.p.e. cancels out all other energies" thing:
http://www.livescience.com/33129-total-energy-universe-zero.html
According to my limited GR knowledge, the total energy-momentum in a frame is given by the stress-energy-momentum pseudotensor. A pseudotensor is not a tensor. This means that it will give you different values depending on your coordinate system. As you may know, there is no "preferred" frame of reference for any system, and this applies to the universe. Therefore, calculating the total energy of the universe as a whole becomes impossible. Of course, calculations in a coordinate system will always agree with the total energy energy-momentum calculated in that frame, so CoE will apply (locally), but no such agreement occurs between different frames. I'm no pro on this topic though, and an expert opinion would be welcome :)

 

[haisydinh]

Thanks for your answer, PWiz! However, even after reading through the article, my trouble still remains. I understand that energy can be conserved if we consider gravitational field to be negative; but how does that actually work in a universe that is accelerating due to the dark energy?
The amount of dark energy increases as the universe expands (because energy density of dark energy stays constant, but the volume increases), which means that dark energy is being created and put into the universe at the same time as the universe expands. So my question is: what sort of gravitational field that can cancel out the positive energy stored within this newly-created dark energy? My speculation (as I've mentioned in my first post) is that this gravitational field actually comes from this newly-created dark energy itself; and I'm wondering whether this is correct.

According to my limited GR knowledge, the total energy-momentum in a frame is given by the stress-energy-momentum pseudotensor. A pseudotensor is not a tensor. This means that it will give you different values depending on your coordinate system. As you may know, there is no "preferred" frame of reference for any system, and this applies to the universe. Therefore, calculating the total energy of the universe as a whole becomes impossible. Of course, calculations in a coordinate system will always agree with the total energy energy-momentum calculated in that frame, so CoE will apply (locally), but no such agreement occurs between different frames. I'm no pro on this topic though, and an expert opinion would be welcome :)

Regarding this paragraph of your post, I'm afraid I don't really follow. My understanding of GR is only basic & conceptual, but not mathematical. I still have lots to learn. However, i guess what you're saying here is that it's not possible to know the total energy of the universe. And if that is the case, then there's no point of discussing conservation of energy in GR. This is probably a fair point; however I still don't understand why so many scientists claim that negative energy of gravitational field cancels out positive energy of the newly-created dark energy, thus showing that an accelerating universe doesn't violate the conservation law.

 

[Chalnoth]

Another way to put it: You can make it so that there is a quantity you call "energy" which is conserved in General Relativity by picking a specific choice of coordinates and by limiting yourself to only certain geometries of the universe. But that quantity will not be conserved in other frames, or with other geometries.

Basically this arises from the fact that the conserved quantity in General Relativity is the stress-energy tensor. The stress-energy tensor is an object which contains energy, momentum, pressure, and twisting forces. In many situations, the conservation of this tensor forces energy, pressure, and potentially momentum and twisting forces to all change together in a very specific way. Because the way these quantities change over time is fixed by the conservation of the stress-energy tensor, it is sometimes possible to add a "gravitational potential energy" which, when added to the energy, causes this new total energy to be conserved within that system. It's not always possible, though.

 

[Soul Intent]

I know there has been a lot of threads on the forum about this topic, but my question is slightly different from the others. I have recently read this article by Sean Carroll (http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/); now, I'm completely aware (& convinced) that energy is not conserved in GR, precisely because of the reasons mentioned in that article. However, in that same article, Sean admitted that it is not incorrect to think that energy is conserved in GR, and i'd like to understand why he could say so. Here is the quote:
Now, what does he mean exactly by this quote? Re-reading this several time, my understanding is this: Although the total dark energy increases as the universe expands, this does not violate the energy conservation. This is because the increase in dark energy also causes more curvature in space-time (due to GR), which consequently increases the gravitational field. And since the gravitational field stores negative energy, it cancels out the positive energy of the dark energy.
Is this the correct description of what actually is going on? Or am I just misinterpreting Sean's article? I mean this explanation sounds a bit weird to me.

Thank you very much in advance! :)

Well if energy ultimately is not conserved, then why isn't it incorrect to say that it is conserved?
He is effectively stating that the law of conservation of mass and energy is wrong. I mean, am I missing something?
I do like and I do understand what he's saying, and it's similar to your interpretation as well, but at the same time this is the first I've heard of the notion and I think there is a lot of evidence that makes this law pretty well understood as it stands.
I'm not necessarily opposed to the debate, but how do the implications of what he's preposing manifest in other aspects of the universe and physics?
Maybe I'm missing a bigger picture, but until I can see a demonstration of how this "loss of energy," occurs and understand what the full range of what "non-conservation" means across the board in GR or otherwise, I'm going to stick to my guns with conservation.

 

[Chalnoth]

Energy is conserved in special cases. So as long as you qualify the specific case you're talking about, it is perfectly-valid to say that energy is conserved.

Mass isn't conserved at all in physics (it changes all the time by tiny amounts, though significant changes only occur in high-energy reactions).

 

[Soul Intent]

So while energy can be lost, it's also correct to say it can be conserved? Under specific conditions.

 

[Chalnoth]

So while energy can be lost, it's also correct to say it can be conserved? Under specific conditions.

Or gained.

But yes, the conditions are quite specific. The pedantic way of say it is that if you can write down the Lagrangian for a system, and that Lagrangian is independent of time, then energy will be conserved for that system. As for the Lagrangian, well, that in itself is a big rabbit hole of fascinating physics and math:
http://en.wikipedia.org/wiki/Action_(physics)

 

[PeterDonis]

He is effectively stating that the law of conservation of mass and energy is wrong.

No, he's not. More precisely, he's not saying that the law of local conservation of stress-energy is wrong. It isn't. Locally, the stress-energy tensor is conserved; more precisely, its covariant divergence is zero, which says that stress-energy cannot be created or destroyed in any small region of spacetime. So all your ordinary intuitions about not being able to create or destroy "stuff" are correct: you still can't build perpetual motion machines, you still can't conjure up matter or energy out of nothing, and matter and energy don't just disappear into nowhere.

The issue with energy non-conservation in GR is that there is no unique way to extend the local conservation law into a global conservation law, because there's no unique way to "add up" all the little bits of stress-energy in each little bit of space at a given time into a "total energy". One key reason for that is that there's no unique way to split spacetime up into "space" and "time" so that you can define "space at a given time". In some very special spacetimes, you can find a way of doing this that is "picked out" by the symmetries of the spacetime, and use that to define your "total energy" sum. But in general you can't, and even in many cases where you can, there are different ways of doing it that give different answers.

 

[PWiz]

So while energy can be lost, it's also correct to say it can be conserved? Under specific conditions.

I think the users here have explained the concepts very well. External energy really depends on how you look at it, literally. It is a property that changes depending upon your frame of reference. But no matter how you look, the total energy that you'll calculate from that perspective will remain constant. It's just that changing perspectives (frames actually) changes the total energy value you perceive, but in all cases the total energy that you "see" will stay constant (calculations in a particular frame all agree with each other).

 

[bilal99uzzzz]

For a precise answer i think we should first define energy . not just the ability to do work , but what really energy is .

 

[Chalnoth]

For a precise answer i think we should first define energy . not just the ability to do work , but what really energy is .

Energy is the quantity that is conserved as a result of time symmetry.

 

[stevebd1]

Now, what does he mean exactly by this quote? Re-reading this several time, my understanding is this: Although the total dark energy increases as the universe expands, this does not violate the energy conservation. This is because the increase in dark energy also causes more curvature in space-time (due to GR), which consequently increases the gravitational field. And since the gravitational field stores negative energy, it cancels out the positive energy of the dark energy.
Is this the correct description of what actually is going on? Or am I just misinterpreting Sean's article? I mean this explanation sounds a bit weird to me.

One thing that could be considered is that the equation of state (w) of dark energy is -1 where w=p/(\rho c^2) and as energy density and pressure are synonymous, this could imply that the negative pressure cancels out the positive energy density which means that DE complies with the first law, the total energy of the universe remains zero. The negative pressure is what also gives dark energy it's negative (repulsive) gravity. The basic algebraic equation for Einsteins gravity is g=\rho c^2 + 3p so zero gravity occurs for matter with an equation of state of -1/3 and anything lower would have negative gravity, dark energy has g=-2.

 

[PeterDonis]

this could imply that the negative pressure cancels out the positive energy density

No, it doesn't. Pressure is not the same as energy density.

 

[stevebd1]

While I wouldn't say that pressure and energy density are the same, I would say that they are synonymous-

\text{Pressure}\ (P)=\frac{\text{Force}}{\text{Area}}=\frac{\text{F.d}}{\text{A.d}}=\frac{\text{W}}{\text{V}}=\frac{\text{Energy}}{\text{Volume}}=\text{Energy density}\

source- http://hyperphysics.phy-astr.gsu.edu/hbase/press.html

 

[PeterDonis]

I would say that they are synonymous

They have the same units, but that doesn't make them the same physically.

It is true that, if you do an integral over an isolated object to obtain its mass, the integral will use the stress-energy tensor, not just energy density, so it will include pressure. (This is called the "Komar mass" of the object.) But the combination that will occur is $\rho c^2 + 3 p$, the same one you gave--not $\rho c^2 + p$. (And the Komar mass is not well-defined for the universe as a whole anyway, since it only works in stationary spacetimes and the universe as a whole is not stationary.)

 

[Chalnoth]

The easiest way to see that the idea of pressure balancing energy density to conserve energy doesn't work is to look at other fields. Radiation, for example, has a positive pressure equal to one third its energy density. As the universe expands, the energy density of radiation decreases as the fourth power of the scale factor, and its pressure decreases proportionately.

This means that the energy per comoving volume decreases linearly with expansion.

There is no way to make sense of this as seeing pressure and energy density as balancing one another.

Curiously, though, if you imagine a hollow box that is expanding, and the contents of that box have some pressure, then the change in energy of the stuff inside is equal to and opposite the amount of work that the pressure does on the walls of the box as it expands.

 

[stevebd1]

I did a search for 'dark energy conservation of energy' and thought I would be worth posting some of the results-

Fact or Fiction?: Energy Can Neither Be Created Nor Destroyed
http://www.scientificamerican.com/article/energy-can-neither-be-created-nor-destroyed/

Modern cosmology has offered up new riddles in energy conservation. We now know that the universe is expanding at a faster and faster rate—propelled by something scientists call dark energy. This is thought to be the intrinsic energy per cubic centimeter of empty space. But if the universe is a closed system with a finite amount of energy, how can it spawn more empty space, which must contain more intrinsic energy, without creating additional energy?

It turns out that in Einstein’s theory of general relativity, regions of space with positive energy actually push space outward. As space expands, it releases stored up gravitational potential energy, which converts into the intrinsic energy that fills the newly created volume. So even the expansion of the universe is controlled by the law of energy conservation.

Expansion of the Universe, Dark Energy and Conservation of energy
https://www.physicsforums.com/threa...ark-energy-and-conservation-of-energy.416650/

GR point of view: There is no global conservation in the universe, because the FRW metric is time dependent. In GR there is only local energy conservation enforced by
\nabla_{\mu}T^{\mu\nu}=0
, and adding a term to the stress tensor of the form
\Lambda\,g_{\mu\nu}
doesn't change that.

Newtonian point of view: As the universe gets bigger, there is more energy from dark energy. However, as the universe gets bigger, PdV work is done on it by sources of pressure. Since dark energy has a negative pressure, this work is negative, and so the two energies exactly cancel out to conserve energy.

How is dark energy consistent with conservation of mass and energy?
http://physics.stackexchange.com/qu...nsistent-with-conservation-of-mass-and-energy

..I decided to Google this and came upon this article:

http://scienceblogs.com/startswithabang/2011/12/02/dark-energy-accelerated-expans/

It says that dark energy does NOT violate conservation and quotes Carroll, Press, and Turner (1992):

"…the patch does negative work on its surroundings, because it has negative pressure. Assuming the patch expands adiabatically [i.e. without loss or gain of heat], one may equate this negative work to the increase of mass/energy of the patch. One thereby recovers the correct equation of state for dark energy: P = – ρ c2. So the mathematics is consistent."

(I'm not sure how legit scienceblogs is but having read the article it seemed ok)Regarding the conservation of energy, the expanding universe and redshift, there was this post-

Violation of conservation of energy by expansion of the universe
https://www.physicsforums.com/threa...f-energy-by-expansion-of-the-universe.372877/

Here are multiple ways to think about it.

1) There should be no conversation of energy globally in our universe because there is no timelike Killing vector in the FRW metric. That is, there is nothing we can identify as energy and say that it is conserved.

2) GR automatically forces a kind of energy conservation \nabla_{\mu}T^{\mu\nu} = 0, where T is the energy-momentum tensor. This is enforced by the Bianchi identity and the Einstein field equations.

3) Newtonian perspective: The redshifted light is compensated for by a change in volume. That is, there is a pressure, so PdV work is done to expand the universe, exactly compensating for the redshift. In fact,using Newtonian arguments of energy conservation, you can derive the Friedmann equations, which describe the expansion of the universe.

 

[PWiz]

Isn't this exactly why we rely upon the universal invariance of the cosmological constant? If the universe is expanding, then to maintain the value of the constant, the energy content must increase as well (in the form of dark energy here). This way, you don't have to deal with global energy conservation.